For a group $G$, let $\operatorname{Aut}(G)$ denote the group of group automorphisms of $G$. (The group operation of $\operatorname{Aut}(G)$ is composition.) Let $p$ be a prime number. Show that the multiplicative group $\mathbb{F}_p \setminus \{0\}$ is isomorphic to $\operatorname{Aut}((\mathbb{F}_p, +))$ under the map $a \mapsto [b \mapsto ab]$ ($a \in \mathbb{F}_p \setminus \{0\}$, $b \in \mathbb{F}_p$).